Vector Magnitude and Angle Calculator

Calculator

Formula Used

2D components → magnitude & angle

• $|\\mathbf v| = \\sqrt{x^2 + y^2}$
• $\\theta = \\operatorname{atan2}(y,x)$
• Unit vector $\\hat{\\mathbf v} = \\dfrac{\\mathbf v}{|\\mathbf v|} = \\left(\\dfrac{x}{|\\mathbf v|},\\dfrac{y}{|\\mathbf v|}\\right)$

2D magnitude + angle → components

• $x = |\\mathbf v|\\cos\\theta,\\quad y = |\\mathbf v|\\sin\\theta$

3D components → spherical angles

• $|\\mathbf v| = \\sqrt{x^2 + y^2 + z^2}$
• Azimuth $\\varphi = \\operatorname{atan2}(y,x)$
• Elevation $\\theta = \\operatorname{atan2}\\big(z,\\sqrt{x^2+y^2}\\big)$
• Direction cosines: $\\alpha=\\arccos\\frac{x}{|\\mathbf v|},\\;\\beta=\\arccos\\frac{y}{|\\mathbf v|},\\;\\gamma=\\arccos\\frac{z}{|\\mathbf v|}$
• Unit vector $\\hat{\\mathbf v}=\\big(x,y,z\\big)/|\\mathbf v|$

3D magnitude + angles → components

• Using elevation from the $xy$-plane and azimuth from +x toward +y:
• $x = |\\mathbf v|\\cos\\theta\\cos\\varphi,\\; y = |\\mathbf v|\\cos\\theta\\sin\\varphi,\\; z = |\\mathbf v|\\sin\\theta$

How to Use

1. Select 2D or 3D, and choose the input mode.
2. Enter values. For 3D, azimuth is in the xy-plane.
3. Pick degrees or radians and desired decimal precision.
4. Click Calculate. Review magnitude, angles, unit vector, and components.
5. Use Download CSV or Download PDF to export history.